A dominate first half performance from Reading was quickly undone when Emma Bissell put the visitors ahead just before half time.
The Royals were able to find an equaliser on 50 minutes as Fara Williams found the back of the net.
Tanya Oxtoby's side remain bottom of the Women's Super League table, despite securing their second point of the season.
Reading stay comfortably mid-table as they prepare to face Manchester United next Sunday.
Story of the match
The hosts' next opportunity came a few minutes later when Danielle Carter was through on goal but a well-timed challenge from Gemma Evans denied the forward a goal.
City's first chance came just shy of the 20 minute mark as Charlie Wellings fired her effort into the side netting.
Reading continued to dominate and Baggaley was kept busy as she tipped a Natasha Harding shot onto the crossbar on 28 minutes.
There was some concern for City moments after as Ebony Salmon went down and required some treatment, but was soon back on for the Robins.
It was Oxtoby's side, though, that found an opener just before half-time.
Moments earlier, Baggaley denied another Reading chance and City worked it up the other end and Wellings found Bissell in the box, who slotted it past Grace Moloney to ensure they took a 1-0 lead into the break.
A goal had been coming for Reading and it didn't take them long to find an equaliser.
Right on the 50 minute mark, Williams finally managed to get the better of Baggaley and send an effort into the back of the net.
The hosts then found their rhythm once again and chance after chance kept coming for Kelly Chambers' side.
10 minutes later, Harding saw her chance to put the Royals ahead glance agonisingly wide of the post, then on 65 minutes Jessica Fishlock had a go from the edge of the box but her shot was pushed wide by the City keeper.
The Robins remained dangerous on the counter attack and nearly saw a well-worked move put them back in the lead on 73 minutes, but Wellings couldn't steer her effort into the back of the net.
Reading came painfully close to making it 2-1 with 13 minutes left to play as Harding found herself one on one with the keeper, but once again her effort was just off target.
The hosts were left frustrated on 90 minutes as they walked away with just a point, but City will be delighted with a resilient defensive display.
Player of the match
As Reading performed for their home crowd, Baggaley kept the visitors well within contention and ultimately earned them the point with a fine goalkeeping display.
Tested constantly across the 90 minutes, the 24-year-old kept her composure and stepped up when needed to ensure the hosts didn't run away with the game.
As City prepare for a huge second half of the season, their goalkeeper will be key to their survival hopes.
Takeaways from the game
Reading lack the final product
The hosts dominated across both halves. The build-up play was there and they were able to work themselves into some really good goalscoring opportunities - but could never quite execute that final ball.
Baggaley was a stand-out performer for the visitors, but the Royals' shots were often just off target and they proved to be wasteful in front of goal.
Williams took the chance when needed, though, and her equaliser was just enough to secure Reading a point.
But it could've, and probably should've been more.
City dangerous on the counter-attack
The visitors spent the majority of the game in their own half, defending against a Reading side being spurred on by their home supporters.
But with patience, chances came and they looked dangerous every time they broke forward - with Salmon leading the line.
Bristol City are next in action on Wednesday night as they host Aston Villa, whilst Reading next face Manchester United in a weeks time.